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Finite-Time Mittag-Leffler Stability of Fractional-Order Quaternion-Valued Memristive Neural Networks with Impulses

  • A. Pratap
  • R. Raja
  • J. Alzabut
  • J. Dianavinnarasi
  • J. CaoEmail author
  • G. Rajchakit
Article
  • 71 Downloads

Abstract

The finite-time Mittag-Leffler stability for fractional-order quaternion-valued memristive neural networks (FQMNNs) with impulsive effect is studied here. A new mathematical expression of the quaternion-value memductance (memristance) is proposed according to the feature of the quaternion-valued memristive and a new class of FQMNNs is designed. In quaternion field, by using the framework of Filippov solutions as well as differential inclusion theoretical analysis, suitable Lyapunov-functional and some fractional inequality techniques, the existence of unique equilibrium point and Mittag-Leffler stability in finite time analysis for considered impulsive FQMNNs have been established with the order \(0<\beta <1\). Then, for the fractional order \(\beta \) satisfying \(1<\beta <2\) and by ignoring the impulsive effects, a new sufficient criterion are given to ensure the finite time stability of considered new FQMNNs system by the employment of Laplace transform, Mittag-Leffler function and generalized Gronwall inequality. Furthermore, the asymptotic stability of such system with order \(1<\beta <2\) have been investigated. Ultimately, the accuracy and validity of obtained finite time stability criteria are supported by two numerical examples.

Keywords

Quaternion-valued Memristor Fractional-order neural networks Finite-time stability 

Notes

References

  1. 1.
    Song C, Fei S, Cao J, Huang C (2019) Robust synchronization of fractional-order uncertain chaotic systems based on output feedback sliding mode control. Mathematics 7(7):599.  https://doi.org/10.3390/math7070599 CrossRefGoogle Scholar
  2. 2.
    Huang C, Zhao X, Wang X, Wang Z, Xiao M, Cao J (2019) Disparate delays-induced bifurcations in a fractional-order neural network. J Franklin Inst 356(5):2825–2846MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Huang C, Cao J (2018) Impact of leakage delay on bifurcation in high-order fractional BAM neural networks. Neural Netw 98:223–235CrossRefGoogle Scholar
  4. 4.
    Rajchakit G, Pratap A, Raja R, Cao J, Alzabut J, Huang C (2019) Hybrid control scheme for projective lag synchronization of Riemann–Liouville sense fractional order memristive BAM neural networks with mixed delays. Mathematics 7(8):759.  https://doi.org/10.3390/math7080759 CrossRefGoogle Scholar
  5. 5.
    Chen B, Chen J (2016) Global \(O(t^{-\alpha })\) stability and global asymptotical periodicity for a non-autonomous fractional-order neural networks with time-varying delays. Neural Netw 73:47–57zbMATHCrossRefGoogle Scholar
  6. 6.
    Huang C, Long X, Huang L, Fu S (2019) Stability of almost periodic Nicholson’s blowflies model involving patch structure and mortality terms. Can Math Bull.  https://doi.org/10.4153/S0008439519000511 CrossRefGoogle Scholar
  7. 7.
    Long X, Gong S (2019) New results on stability of Nicholson’s blowflies equation with multiple pairs of time-varying delays. Appl Math Lett.  https://doi.org/10.1016/j.aml.2019.106027 CrossRefGoogle Scholar
  8. 8.
    Liu Y, Tong L, Lou J, Lu J, Cao J (2017) Sampled-data control for the synchronization of Boolean control networks. IEEE Trans Cybern.  https://doi.org/10.1109/TCYB.2017.2779781 CrossRefGoogle Scholar
  9. 9.
    Huang C, Qiao Y, Huang L, Agarwal RP (2018) Dynamical behaviors of a food-chain model with stage structure and time delays. Adv Differ Equ 2018:186.  https://doi.org/10.1186/s13662-018-1589-8 MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Huang C, Zhang H, Huang L (2019) Almost periodicity analysis for a delayed Nicholson’s blowflies model with nonlinear density-dependent mortality term. Commun Pure Appl Anal 18(6):3337–3349MathSciNetCrossRefGoogle Scholar
  11. 11.
    Huang C, Zhang H, Cao J, Hu H (2019) Stability and Hopf bifurcation of a delayed prey-predator model with disease in the predator. Int J Bifurc Chaos 29(7):1950091 23 PagesMathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Huang C, Yang Z, Yi T, Zou X (2014) On the basins of attraction for a class of delay differential equations with non-monotone bistable nonlinearities. J Differ Equ 256(7):2101–2114MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Rakkiyappan R, Sivaranjani R, Velmurugan G, Cao J (2016) Analysis of global \(O(t^{-\alpha })\) stability and global asymptotical periodicity for a class of fractional-order complex-valued neural networks with time varying delays. Neural Netw 77:51–69zbMATHCrossRefGoogle Scholar
  14. 14.
    Chen J, Li C, Yang X (2018) Asymptotic stability of delayed fractional-order fuzzy neural networks with impulse effects. J Franklin Inst 355(15):7595–7608MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Rakkiyappan R, Velmurugan G, Cao J (2015) Stability analysis of fractional-order complex-valued neural networks with time delays. Chaos Solitons Fractals 78:297–316MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Liu P, Nie X, Liang J, Cao J (2018) Multiple Mittag-Leffler stability of fractional-order competitive neural networks with Gaussian activation functions. Neural Netw 108:452–465CrossRefGoogle Scholar
  17. 17.
    Yang X, Song Q, Liu Y, Zhao Z (2015) Finite-time stability analysis of fractional-order neural networks with delay. Neurocomputing 152:19–26CrossRefGoogle Scholar
  18. 18.
    Wu R, Hei X, Chen L (2013) Finite-Time stability of fractional-order neural networks with delay. Commun Theor Phys 60:189–193MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Chua L (1971) Memristor: the missing circuit element. IEEE Trans Circuits Theory 18:507–519CrossRefGoogle Scholar
  20. 20.
    Bernard W (1960) An adaptive adaline neuron using chemical memistors, Stanford Electronics Laboratories Technical Report 1553-2Google Scholar
  21. 21.
    Strukov D, Snider G, Stewart D, Williams R (2008) The missing memristor found. Nature 453:80–83CrossRefGoogle Scholar
  22. 22.
    Kim H, Sah M, Yang C, Roska T, Chua L (2012) Memristor bridge synapses. Proc IEEE 100(6):2061–2070.  https://doi.org/10.1109/JPROC.2011.2166749 [6074916]CrossRefGoogle Scholar
  23. 23.
    Miller K, Nalwa K, Bergerud A, Neihart N, Chaudhary S (2010) Memristive behavior in thin anodic titania. IEEE Electron Device Lett 31(7):737–739CrossRefGoogle Scholar
  24. 24.
    Sun J, Shen Y, Yin Q, Xu C (2012) Compound synchronization of four memristor chaotic oscillator systems and secure communication. Chaos 22(4):1–10zbMATHGoogle Scholar
  25. 25.
    Corinto F, Ascoli A, Gilli M (2011) Nonlinear dynamics of memristor oscillators. IEEE Trans Circuits Syst I 58(6):1323–1336MathSciNetCrossRefGoogle Scholar
  26. 26.
    Wang L, Song Q, Liu R, Zhao Z, Alsaadi F (2017) Finite-time stability analysis of fractional-order complex-valued memristor-based neural networks with both leakage and time-varying delays. Neurocomputing 245:86–101CrossRefGoogle Scholar
  27. 27.
    Rakkiyappan R, Velmurugan G, Cao J (2014) Finite-time stability analysis of fractional-order complex-valued memristor-based neural networks with time delays. Nonlinear Dyn 78:2823–2836MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Sudbery A (1979) Quaternionic analysis, Math. Proc. Camb. Phil. SocGoogle Scholar
  29. 29.
    Hu J, Zeng C, Tan J (2017) Boundedness and periodicity for linear threshold discrete-time quaternion-valued neural network with time-delays. Neurocomputing 267:417–425CrossRefGoogle Scholar
  30. 30.
    Tu Z, Cao J, Ahmed A, Hayat T (2017) Global dissipativity analysis for delayed quaternion-valued neural networks. Neural Netw 89:97–104CrossRefGoogle Scholar
  31. 31.
    Tu Z, Zhao Y, Ding N, Feng N, Wei Z (2019) Stability analysis of quaternion-valued neural networks with both discrete and distributed delays. Appl Math Comput 343:342–353MathSciNetGoogle Scholar
  32. 32.
    Tan M, Liu Y, Xu D (2019) Multistability analysis of delayed quaternion-valued neural networks with nonmonotonic piecewise nonlinear activation functions. Appl Math Comput 341:229–255MathSciNetGoogle Scholar
  33. 33.
    You X, Song Q, Liang J, Liu Y, Alsaadi F (2018) Global \(\mu \)-stability of quaternion-valued neural networks with mixed time-varying delays. Neurocomputing 290:12–25CrossRefGoogle Scholar
  34. 34.
    Samidurai R, Sriraman R, Zhu S (2019) Leakage delay-dependent stability analysis for complex-valued neural networks with discrete and distributed time-varying delays. Neurocomputing 338:262–273CrossRefGoogle Scholar
  35. 35.
    Samidurai R, Sriraman R, Cao J, Tu Z (2018) Effects of leakage delay on global asymptotic stability of complex-valued neural networks with interval time-varying delays via new complex-valued Jensens inequality. Int J Adapt Control Signal Process 32:1294–1312MathSciNetzbMATHGoogle Scholar
  36. 36.
    Tan M, Liu Y, Xu D (2019) Multistability analysis of delayed quaternion valued neural networks with nonmonotonic piecewise nonlinear activation functions. Appl Math Comput 341:229–255MathSciNetGoogle Scholar
  37. 37.
    Zhou Y, Li C, Chen L, Huang T (2018) Global exponential stability of memristive Cohen–Grossberg neural networks with mixed delays and impulse time window. Neurocomputing 275:2384–2391CrossRefGoogle Scholar
  38. 38.
    Ning L, Cao J (2018) Global dissipativity analysis of quaternion-valued memristor-based neural networks with proportional delay. Neurocomputing 321:103–113CrossRefGoogle Scholar
  39. 39.
    Liu Y, Zheng Y, Lu J, Cao J, Rutkowski L (2019) Constrained quaternion-variable convex optimization: a quaternion-valued recurrent neural network approach. IEEE Trans Neural Netw Learn Syst.  https://doi.org/10.1109/TNNLS.2019.2916597 CrossRefGoogle Scholar
  40. 40.
    Yang X, Li C, Song Q, Chen J, Huang J (2018) Global Mittag-Leffler stability and synchronization analysis of fractional-order quaternion-valued neural networks with linear threshold neurons. Neural Netw 105:88–103CrossRefGoogle Scholar
  41. 41.
    Xiao J, Zhong S (2019) Synchronization and stability of delayed fractional-order memristive quaternion-valued neural networks with parameter uncertainties. Neurocomputing 363:321–338CrossRefGoogle Scholar
  42. 42.
    Li X, Ding Y (2017) Razumikhin-type theorems for time-delay systems with persistent impulses. Syst Control Lett 107:22–27MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Yang X, Li X, Xi Q, Duan P (2018) Review of stability and stabilization for impulsive delayed systems. Math Biosci Eng 15(6):1495–1515MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Li X, Ho DWC, Cao J (2019) Finite-time stability and settling-time estimation of nonlinear impulsive systems. Automatica 99:361–368MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Wu H, Zhang X, Xue S, Wang L, Wang Y (2016) LMI conditions to global Mittag-Leffler stability of fractional-order neural networks with impulses. Neurocomputing 193:148–154CrossRefGoogle Scholar
  46. 46.
    Zhang X, Niu P, Ma Y, Wei Y, Li Guoqiang (2017) Global Mittag-Leffler stability analysis of fractional-order impulsive neural networks with one-side Lipschitz condition. Neural Netw 94:67–75CrossRefGoogle Scholar
  47. 47.
    Wang L, Song Q, Liu Y, Zhao Z, Alsaadi F (2017) Global asymptotic stability of impulsive fractional-order complex-valued neural networks with time delay. Neurocomputing 243:49–59CrossRefGoogle Scholar
  48. 48.
    Podlubny I (1999) Fractional differential equations. Academic Press, San DiegozbMATHGoogle Scholar
  49. 49.
    Ye H, Gao J, Ding Y (2007) A generalized Gronwall inequality and its application to a fractional differential equation. J Math Anal Appl 328(2):1075–1081MathSciNetzbMATHCrossRefGoogle Scholar
  50. 50.
    Zhang S, Yu Y, Wang H (2015) Mittag-Leffler stability of fractional-order Hopfield neural networks. Nonlinear Anal Hybrid Syst 16:104–121MathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    De la Sen M (2011) About robust stability of Caputo linear fractional dynamic systems with time delays through fixed point theory, Fixed Point Theory and Applications (1), Article ID: 867932, 1-19Google Scholar
  52. 52.
    Chen B, Chen J (2015) Global asymptotical \(\omega \)-periodicity of a fractional-order non-autonomous neural networks. Neural Netw 68:78–88zbMATHCrossRefGoogle Scholar
  53. 53.
    Filippov AF (1988) Differential equations with discontinuous right-hand sides. Kluwer, DordrechtzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Vel Tech High Tech Dr Rangarajan Dr Sakunthala Engineering CollegeChennaiIndia
  2. 2.Ramanujan Centre for Higher MathematicsAlagappa UniversityKaraikudiIndia
  3. 3.Department of Mathematics and General SciencesPrince Sultan UniversityRiyadhSaudi Arabia
  4. 4.School of MathematicsSoutheast UniversityNanjingChina
  5. 5.Department of Mathematics, Faculty of ScienceMaejo UniversityChiang MaiThailand

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